London and Paris in October

Museum would be no problem in either London or Paris as they are open all year.

What estates are you looking at in particular?

We were in London the last 2 weeks of October last year and discovered it was half-term. In other words kids were out of school. We didn’t find that things were closed there, nor have they been the two times I’ve been to Paris in October. What private English estates are you wanting to see?

There are no Covid restrictions in London or the UK.

I don't know of any museum that closes during the year except for certain major holidays like Christmas.

Could you give examples of what you found closed? It's possible that what you saw was a seasonal schedule change and not a closure. They may have not put up their winter schedules yet.

Many of the English stately homes close after the School half term at the end of October, then re-open for Christmas and then close again until close to Easter. But it would be rare to find any that close earlier than end October. But major museums in big cities open all year. Smaller museums in provincial towns often close for winter.

Lots of great museums in those cities, but let me warn you about kids that age and museums, they are ready to leave after 30 minutes.
My 8 year old commented about all the "naked statutes" in the Greek and Roman section of the Louvre.

So you are traveling to two of the most popular cities on the planet and you wish to avoid crowds.

After the Covid lockdown people are traveling again. There will be crowds even in October. I travel to London nearly every November. Its crowded. Not massive hordes like in the Summer months but tourists abound.

If you select London consider the following places for the kids:
Mudchute Farm
Tower Bridge Experience
Hamleys Toy Shop
Visit The Princess Diana playground in Kensington Gardens
Harrods Food Halls
Horniman Museum
Take them to Transport Museum
Legoland in Windsor
Ride upstairs in the front on a Boris bus
Watching the Changing of the Horse Guards on the Horse Guards parade then meander through St James Park
Watch the feeding of the pelicans in St James park
Walk through Richmond Park hoping to see the fallow deer

We don’t want to travel when things are crowded

But you are going to London and Paris. They are always crowded.

How many days do you plan to be on the ground in Europe (in other words, not counting flying days)? By dividing your time between not just two cities, but two countries with different currencies and different languages, you'll be doubling the potential stress and the effort/energy it will take. If you have a full two weeks, that will probably work out all right, but anything less than that will make the trip pretty frantic, IMO. Also, in October the weather may be chilly and rainy, limiting outdoor activities and even making it a chore to walk around on city streets from one museum or place of interest to the next. You'll definitely want layers of clothing, raincoats, and shoes that are either waterproof or treated with a waterproofing product before you leave home.

We have done both Paris and London with our son (Paris when he was 4, London when he was 15 months and again this past fall, when he was 5). Both destinations are excellent for kids, and for first-timers to Europe. However, I would caution against trying to pack too much into a "short stay". If you have a week or less (7 nights in the country, not including your overnight flight and your departure day), then I would pick ONE. Taking the Eurostar between London and Paris is easy, but will eat up a full day.

I'm not sure what "private English estates" you'd be looking at in London. We've visited small, privately-owned estates and halls out in the English countryside, but there aren't any private estates (as I understand it - I'm more than happy to be corrected) in London. Tourist sites in London and Paris will be open and generally aren't affected by an "offseason", because there generally isn't one in those cities.

Although you won't be able to eliminate the crowds, there are certainly ways that you can manage them (by booking tickets ahead, arriving right at opening time, etc.) and have an enjoyable trip regardless.